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Chapter 11: Problem 41

Find the vertex of the graph of each function. $$ g(x)=(x+2)^{2} $$

Short Answer

Expert verified
The vertex is (-2, 0).

Step by step solution

01

Identify the function form

The function given is \( g(x) = (x+2)^2 \). This is in the form \( (x-h)^2 + k \), which is a standard form of a quadratic function representing a parabola. Here, \( h \) and \( k \) are the coordinates of the vertex.
02

Determine the vertex coordinates

The vertex form of a quadratic function is \( (x-h)^2 + k \), where \( h \) is a horizontal shift and \( k \) is a vertical shift. For the function \( g(x) = (x+2)^2 \), we have \( h = -2 \) and \( k = 0 \). So, the vertex of the parabola is at \( (h, k) = (-2, 0) \).
03

Confirm the parabola's orientation

Since the function is \( (x+2)^2 \) and there is no negative sign in front, the parabola opens upwards. This confirms that (-2, 0) is indeed the vertex.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Function
A quadratic function is a type of polynomial function where the highest degree of the variable is two. It is usually expressed in the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. Quadratic functions create curves called parabolas when graphed on a coordinate plane.

These functions are important in various fields of science and engineering because they model real-world scenarios, such as the trajectory of a ball or the shape of satellite dishes. In our example, the quadratic function given is \( g(x) = (x+2)^2 \), indicating it forms a parabola.
Parabola
A parabola is the U-shaped graph of a quadratic function. It can either open upwards or downwards depending on the leading coefficient of the quadratic term. In general, if \( a > 0 \), the parabola opens upwards, and if \( a < 0 \), it opens downwards.

The parabola is symmetrical about its vertex, which is a key feature. The vertex is either the lowest or highest point on the graph, depending on the direction of the parabola. For the function \( g(x) = (x+2)^2 \), the parabola opens upwards because the coefficient of \( x^2 \) is positive. Thus, it has a minimum at the vertex.
Vertex Form of a Quadratic Function
The vertex form of a quadratic function provides a straightforward way to identify the vertex of the parabola. It is expressed as \( (x-h)^2 + k \), where \( (h, k) \) are the coordinates of the vertex.

This form makes it easy to read the vertex directly from the equation. In the function \( g(x) = (x+2)^2 \), you can rewrite it in the vertex form as \( (x - (-2))^2 + 0 \), indicating that the vertex is at \((-2, 0)\). The vertex form is particularly useful for graphing parabolas because it highlights the shifts from the origin.
Horizontal Shift
A horizontal shift occurs when the graph of the function moves left or right from its original position. In the vertex form \( (x-h)^2 + k \), the value of \( h \) determines the direction and magnitude of the horizontal shift.

If \( h \) is positive, the graph shifts to the right, and if \( h \) is negative, the graph shifts to the left. In the function \( g(x) = (x+2)^2 \), \( h = -2 \). This means the parabola shifts 2 units to the left from the standard position at the origin.
Vertical Shift
A vertical shift involves moving the graph of a function up or down in the coordinate plane. In the vertex form \( (x-h)^2 + k \), \( k \) denotes the vertical shift.

If \( k \) is positive, the graph moves upwards, and if \( k \) is negative, the graph moves downwards. In the equation \( g(x) = (x+2)^2 \), \( k = 0 \), so there is no vertical shift. The vertex begins on the x-axis as there's no vertical displacement. Understanding both horizontal and vertical shifts supports comprehending more complex transformations efficiently.

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Most popular questions from this chapter

Use the quadratic formula to solve each equation. These equations have real number solutions only. $$ \frac{1}{2} x^{2}-x-1=0 $$

Use the discriminant to determine the number and types of solutions of each equation. $$ 4 x^{2}+12 x=-9 $$

Use the discriminant to determine the number and types of solutions of each equation. $$ x^{2}-7=0 $$

Use the quadratic formula to solve each equation. These equations have real number solutions only. $$ x^{2}-6 x+9=0 $$

Solve each equation. $$ \sqrt{y+2}+7=12 $$
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